Verification of shielding effect predictions for large area field emitters

A recent analytical model for large area field emitters [D. Biswas and R. Rudra, Phys. Plasmas 25, 083105 (2018)], based on the line charge model (LCM), provides a simple approximate formula for the field enhancement on hemiellipsoidal emitter tips in terms of the ratio of emitter height to pairwise distance between neighboring emitters. The formula, verified against the exact solution of the linear LCM, was found to be adequate, provided that the mean separation between emitters is larger than half the emitter height, h. In this paper, we subject the analytical predictions to a more stringent test by simulating (i) an infinite regular array and (ii) an isolated cluster of 10 random emitters, using the finite element software COMSOL v5.4. In the case of the array, the error in the apex field enhancement factor (AFEF) is found to be less than 0.25% for an infinite array when the lattice constant c ≥ 1.5h, increasing to 2.9% for c = h and 8.1% for c = 0.75h. For an isolated random cluster of 10 emitters, the error in large AFEF values is found to be small. Thus, the error in the net emitted current is small for a random cluster compared to a regular infinite array with the same (mean) spacing. The LCM thus provides a reasonable analytical tool for optimizing a large area field emitter.